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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 49
Capitolo 1, Problema 49

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.


ƒ o G

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1
Identify the functions involved: \( f(x) = |x| \) and \( G(x) = \frac{1}{x-2} \).
Understand the composition \( (f \circ G)(x) = f(G(x)) \), which means substituting \( G(x) \) into \( f(x) \).
Substitute \( G(x) = \frac{1}{x-2} \) into \( f(x) \), resulting in \( f\left(\frac{1}{x-2}\right) = \left| \frac{1}{x-2} \right| \).
Determine the domain of \( G(x) = \frac{1}{x-2} \), which is all real numbers except \( x = 2 \) because the denominator cannot be zero.
Since \( f(x) = |x| \) is defined for all real numbers, the domain of \( f \circ G \) is the same as the domain of \( G(x) \), which is all real numbers except \( x = 2 \).

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Composite Functions

A composite function is formed when one function is applied to the result of another function. It is denoted as (f o g)(x) = f(g(x)). Understanding how to combine functions is essential for solving problems involving multiple functions, as it requires evaluating the inner function first before applying the outer function.
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Evaluate Composite Functions - Special Cases

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with composite functions, it is crucial to determine the domain of both the inner and outer functions, as the overall domain will be restricted by any values that make either function undefined.
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Finding the Domain and Range of a Graph

Absolute Value Function

The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x regardless of its sign. This function is important in composite functions because it affects the output based on the input's sign, which can influence the overall behavior and domain of the composite function.
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Initial Value Problems
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Domanda del libro di testo

Solve the following equations.


{Use of Tech} sin2θ=15,0<θ<π2\(\sin\)2\(\theta\)=\(\frac\)15,0\(\lt{\theta}\)\(\lt{\frac{\pi}{2}\)}

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Domanda del libro di testo

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


a. Find A(2) .

ƒ(t) =6 <IMAGE>

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Domanda del libro di testo

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


c. Find a formula for A(x)


ƒ(t) =6 <IMAGE>

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Domanda del libro di testo

More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.


ƒ o g

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Domanda del libro di testo

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.



b. Find A(6).



ƒ(t) =6 <IMAGE>

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Domanda del libro di testo

Slope functions Determine the slope function S (x) for the following functions

ƒ(x)=∣x∣ƒ(x) = | x |

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